Kobayashi hyperbolicity in Riemannian manifolds

TL;DR

This paper investigates the boundary behavior of the Kobayashi-Royden metric and hyperbolicity in Riemannian manifolds, establishing new theorems on harmonic discs and providing examples of hyperbolic manifolds.

Contribution

It introduces new results on Kobayashi hyperbolicity, including a Fatou type theorem and a Picard theorem for conformal harmonic discs in Riemannian manifolds.

Findings

Proved a Fatou type theorem for conformal harmonic discs.

Established a Picard theorem for conformal harmonic discs.

Provided examples of Kobayashi hyperbolic Riemannian manifolds.

Abstract

We study the boundary behavior of the Kobayashi-Royden metric and the Kobayashi hyperbolicity of domains in Riemannian manifolds. As an application, we prove a Fatou type theorem on the existence, almost everywhere, of non tangential limits for bounded conformal harmonic immersed discs. We also prove a Picard theorem for conformal harmonic discs and give some examples of Kobayashi hyperbolic Riemannian manifolds.